1993/10/07 by Subhashis Nag, Dennis Sullivan, Nag, Subhashis +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Harmonic Analysis Research #Analytic and geometric function theory #Mathematics and Applications #alg-geom #funct-an #hep-th #math.AG #math.FA
paper · pdf · doi:10.48550/arxiv.alg-geom/9310005
39 pages (TEX)
arxiv created 1993/10/08 · arxiv updated 2009/11/30
The Universal Teichmüller Space, T(1), is a universal parameter space for all Riemann surfaces. In earlier work of the first author it was shown that one can canonically associate infinite- dimensional period matrices to the coadjoint orbit manifold Diff(S1)/Mobius(S1) -- which resides within T(1) as the (Kirillov-Kostant) submanifold of ``smooth points'' of T(1). We now extend that period mapping Π to the entire Universal Teichmüller space utilizing the theory of the Sobolev space H1/2(S1). Π is an equivariant injective holomorphic immersion of T(1) into Universal Siegel Space, and we describe the Schottky locus utilizing Connes' ``quantum calculus''. There are connections to string theory. Universal Teichmüller Space contains also the separable complex submanifold T(H_∞) -- the Teichmüller space of the universal hyperbolic lamination. Genus-independent constructions like the universal period mapping proceed naturally to live on this completion of the classical Teichmüller spaces. We show that T(H_∞) carries a natural convergent Weil-Petersson pairing.